B Sabine Hossenfelder and Beauty in Physics

[PeterDonis]

Neither, I think. The modern language of Weyl fields wasn't popular back then (Griffiths specifically warns against it) and the (1+γ5) terms were viewed as projection operators rather than fields in their own right. That is, the W coupled to the left-handed components of fields rather than left-handed fields. In this view, there is (or at least can be) a non-interacting right-handed component to the neutrino.

Ok, thanks, this is good background that I wasn't aware of.

 

[martinbn]

I tried to read her book but I couldn't. It was a bit boring and I had the constant impression that she tries a bit too hard to be witty all the time, which seemed to me forced and was very tiring. I am not saying that the book isn't worth it, it's just not for me. I also still don't know what the main thesis is.

 

[Android Neox]

From the article:https://motherboard.vice.com/en_us/...ists-are-misled-by-outdated-notions-of-beauty

I wouldn't call multiverse interpretation "beautiful". Everett was kicked out of physics for proposing Many Worlds. The only reason multiverse models exist is because they are the only models that are consistent with quantum entanglement, Bell's Inequality, and non-simultaneity.

 

[Lish Lash]

I would argue that "beauty" is such a culturally loaded term that it is inappropriate to apply it to a field as rigorously objective as physics. Beauty is not only an intrinsically subjective term, it is entangled with intimately personal associations of pleasure and sexual attraction, activities often dismissed as inessential. It seems covertly provocative to apply a term with such connotations to intellectual pursuits of knowledge. I think "elegance" is a more genuine characterization of the esthetic inclinations of scientific intuition.

 

[fresh_42]

I would argue that "beauty" is such a culturally loaded term that it is inappropriate to apply it to a field as rigorously objective as physics. Beauty is not only an intrinsically subjective term, it is entangled with intimately personal associations of pleasure and sexual attraction, activities often dismissed as inessential. It seems covertly provocative to apply a term with such connotations to intellectual pursuits of knowledge. I think "elegance" is a more genuine characterization of the esthetic inclinations of scientific intuition.

This is quibbling. I don't think that beauty is a subjective term. There is such a thing as classic beauty, and you know it if you see it. The point is that uneducated people use this term as well, and for them it is highly subjective and emotional, but this is not the beauty scientists speak of. Why does Beethoven's 9-th work on the entire globe, or Chopin? Why are Michelangelo's sculptures beautiful? These are in the category which is meant here, not someone's favorite color, food or body shape.

Whether you call this elegance is just a workaround of the fact, that many people simply don't have a sense of beauty. You may call this sad, but I think it is true. If scientists speak of beauty, they usually mean the incredible simplicity of many natural phenomenons, or how slim some proofs are, or how a few equations reveal deep insights. There is an inherent beauty in most theories and it can neither be discussed, nor is it a matter of opinion. Galois' theory is beautiful. The only question about it is whether people are able to see it or not. If they don't, then the reason is a lack of understanding, not a lack of beauty.

 

[PeterDonis]

I don't think that beauty is a subjective term. There is such a thing as classic beauty, and you know it if you see it.

I think you're on very shaky ground here. First, "you know it if you see it" is pretty much the classic definition of "subjective". Second, I'm not sure how much agreement there is among scientists and mathematicians about what specific theories, models, proofs, etc. count as "beautiful".

The only question about it is whether people are able to see it or not.

But if they can't see it, and it's defined as "you know it if you see it", how can it possibly be objective?

 

[fresh_42]

Yes, 'You know it if you see it' might have been too bold, as my definition constraints the set of people I mean by 'you'. It is a rather snobbish and certainly political incorrect way to see it, because I couple it to education. However, it is not shaky ground, as I don't want to defend it, neither make it mass compatible. It is my personal opinion, and I declared it as such. It doesn't demand to be 'provable' as I don't have the desire to convince people.

I cannot run the 100 meter under, say 20 seconds, so I am not an athlete. The beauty of science can't be seen by the majority of people, but this doesn't mean there wasn't athletes who can run the 100 meter under 10 seconds.

 

[Lish Lash]

If scientists speak of beauty, they usually mean the incredible simplicity of many natural phenomenons, or how slim some proofs are, or how a few equations reveal deep insights.

While I agree with your characterization of scientists, those are examples of what I (and I believe most dictionaries) would describe as "elegance". You may insist on conflating that term with more fashionably publicized forms of "beauty", but it's a distinction I think we do recognize when we see it.

 

[PeterDonis]

'You know it if you see it' might have been too bold

My issue isn't that it is "too bold", it's that it is subjective, when you specifically tried to claim it wasn't.

If, for example, you argued that General Relativity is "beautiful" because it takes a very simple field equation derived from a very simple Lagrangian and extracts from it a huge variety of detailed quantitative predictions that have been confirmed experimentally to many decimal places, that would be an objective criterion for "beauty" (and one I would tend to agree with). But that criterion, or anything like it, is very different from "you know it when you see it".

For the criterion I just described, I would agree with @Lish Lash that "elegance" is a better word for it than "beauty", but that's a matter of personal preference; the criterion itself is objective. The problem with "you know it when you see it" is that it isn't.

The beauty of science can't be seen by the majority of people, but this doesn't mean there wasn't athletes who can run the 100 meter under 10 seconds.

And if the criterion of "beauty in science" is something like what I described above, there is no problem. Certainly the fraction of people who can understand why GR is beautiful in the above sense is small, just as the fraction of people who can run 100 meters in under 10 seconds is small, but that doesn't make either one any less true or objective.

The problem with "you know it when you see it" is that if you then turn around and say that, well, some people just can't see it, you are undermining your own criterion; it becomes useless.

 

[fresh_42]

The problem with "you know it when you see it" is that if you then turn around and say that, well, some people just can't see it, you are undermining your own criterion; it becomes useless.

Maybe "bold" was the wrong word. It was what the dictionary gave me for the word I was looking for. "You know it if you see it" shouldn't be the definition, rather emphasizing that people who can see it will recognize it as such, in the same sense as people around the world apparently agree that the final chorus in Bethoven's 9-th is beautiful. There is no objective criterion either, nevertheless there are dozens of flash mob videos from around the world which support this point of view.

And to find GR (or my example Galois' theory) beautiful, you first have to understand it. I don't think that this sense of beauty can be measured in bytes as in CS: the shorter the more beautiful. This would be too short in either meaning. And that's my difficulty here and why I only expressed my personal point of view: there is no objective scale, but this does not automatically make it subjective. It depends on whether you're able to see it, i.e. whether you really understood GR. GR is beautiful despite the fact that the rubber sheet isn't accurate. It's beauty doesn't come from the numbers 21 or 18 (I don't remember the exact number of equations, resp. free parameters, which is why I've chosen Galois theory as example).

 

[PeterDonis]

"You know it if you see it" shouldn't be the definition, rather emphasizing that people who can see it will recognize it as such

Ok, fair enough.

It depends on whether you're able to see it, i.e. whether you really understood GR.

Yes, but there are objective criteria for whether people have really understood GR. Can they calculate its predictions for themselves, and have their calculations match the ones that GR experts have already done and which have been extensively confirmed by experiment?

You are basically saying that anyone who meets those objective criteria will find GR beautiful. I'm not sure that's true--at least, I'm not as sure as you seem to be. But I think that it's more likely that anyone who meets those objective criteria will agree that GR is simple and elegant, in the sense I described in my previous post.

there is no objective scale, but this does not automatically make it subjective

I agree. There is no objective scale for "simple and elegant" as I described it, yes. And that does not automatically make that criterion subjective. I think the key is that the criterion has to involve something that can be reasonably objectively measured and agreed on--even if people can't agree on where exactly the boundary lines of categories like "simple and elegant" or "beautiful" are in terms of the objective measures, we can all agree on what the objective measures themselves say in particular cases.

For example, we could argue over how simple a Lagrangian needs to be to qualify as "simple and elegant", but we can all agree, for example, that the Einstein-Hilbert GR Lagrangian is the unique one that has no higher than second derivatives of the metric and minimal coupling between gravity and matter. Similarly, we could argue over how "simple" Galois theory is, but we can all agree that it provides simple insights that resolve a lot of questions (such as why fifth degree and higher polynomials aren't solvable using radicals) that seem intractable without it.

 

[fresh_42]

You are basically saying that anyone who meets those objective criteria will find GR beautiful. I'm not sure that's true--at least, I'm not as sure as you seem to be.

Yes, but this is as if you asked whether a rose is beautiful. I'm not especially a fan of roses, but I accept them as beautiful. I won't discuss this judgement. Anybody who has a different opinion is - to me - just an ignorant. Since I don't want to spread this view, I don't have a problem with people of different opinion. I simply don't take them serious about this topic. Again, a personal attitude, no definition of a hidden truth of the concept of beauty. It is my personal definition.

I remember a talk at a dinner table about the old question: invented or discovered. To me, these two subjects are related, as they come down to the question: what is nature and what is human? I admit to be a Platonist here. I said then as an example, that - again in my opinion - all these things are discovered, even music. I mentioned Tchaikovsky's 6-th as an example, where music tells a story (about death) which everybody can understand, so it cannot be invented. I'm not sure how many scientists would follow this strict point of view, but at least one mathemarician understood what I meant. From that and similar discussions, e.g. about applied versus abstract mathematics, I conclude that at least many scientists share the same awareness of beauty.

Another example, I think, is the standard model. Its simplicity, beside its reliability, seems to be so convincing, that many, if not most attempts to find a unifying model are simply extensions of it: bigger groups (dimensions), graded Lie algebras.

At the hearts of our models is usually Noether and we find it difficult to leave the concept of Lagrangians. To me this is a sign of an inherent beauty, and it can be seen once we dug deep enough to this core. A certain Lagrangian might be ugly, the principle is not.

 

[stevendaryl]

I think you're on very shaky ground here. First, "you know it if you see it" is pretty much the classic definition of "subjective". Second, I'm not sure how much agreement there is among scientists and mathematicians about what specific theories, models, proofs, etc. count as "beautiful".

But if they can't see it, and it's defined as "you know it if you see it", how can it possibly be objective?

I'm not sure that I agree with the claim that beauty is objective, but the fact that not everyone can detect a quality doesn't mean that the quality is subjective. For example, those with red/green color blindness can't tell the difference between red apples and green apples, but that doesn't mean that the difference is subjective.

 

[PeterDonis]

the fact that not everyone can detect a quality doesn't mean that the quality is subjective

I agree with this as a general statement. But if the specific definition of the quality is "you know it when you see it", then if someone doesn't see it, it isn't there--i.e., it's subjective.

However, @fresh_42 has clarified that he didn't mean "you know it when you see it" as a definition, but as an illustration; his reasons for thinking that "beauty" by his definition is objective actually seem pretty similar to my reasons for thinking that "simple and elegant" is objective. See posts #60, #61, and #62.

 

[Auto-Didact]

I also think the very premise is unfair. If Hossenfelder has a better approach, she should publish it. But "you're doing it wrong! You're doing it wrong!" is not helpful.

Actually saying 'stop doing it' is helpful in itself, especially in the chosen formats of public articles and popular books. The argument is that theoretical physics in foundations has not produced any spectacularly successful work after the Standard Model. The duration of this stagnation has even been dubbed a 'crisis in physics' precisely because this has never before occurred in theoretical physics for such a length of time since Newton; the problem is even worse considering there are more physicists alive today than in all of history combined since Newton.

Sociologically speaking, in the last 30 or so years there has been a strong popularising influence in the public domain. Most of this comes from influential scientists promoting MWI, SUSY, the multiverse and so on. This has an effect on the chosen career paths of a significant amount of young science students. Hossenfelder's book, as well as Woit's 'Not Even Wrong' and Smolin's 'The Trouble With Physics', form an opposing voice to unbalanced considerations aesthetic arguments in physics, which has become more and more popularized since Dirac, eventually peaking during the 80s, 90s and 00s. They argue that these unsuccessful theories of physics and their proponents have become dominant in science without experiment and that they continue to have a stable, ideological hold over physics, against which one must actively resist and fight against if theoretical physics is to return its previously healthy state.

Moreover, this has also led to many interested laymen, especially academics and scientists outside physics, to having accepted these models as scientific on the same level as actual canonical theories in physics. These academics, especially the more vocal ones, tend to have influence over university boards and connections across faculties, in other words, some role in selecting who will or will not get hired from the perspective and what ideas will or will not get funded from shared university funds; the recognizability and public familiarity of their ideas plays a significant role in these choices.

I suspect that Hossenfelder et al. realize that the adventure of science is not only an academic endeavor, but because of how science has become institutionalised, it is also a battle for the heart of the public. Given enough weapons and ammo, the public will eventually themselves start questioning theorists more strongly, this actually does occur seeing students are also part of the public; when this happens and these theorists are unable to properly account for why they haven't made any actually large lasting contributions to physics like their predecessors have, things will start to change in a democratic fashion i.e. their funding will get cut and redirected to other competing theories. In other words, science will finally self correct.

It is also important to mention that forcing a student to conform to some PhD topic tends to make him too familiar with some set of techniques and the associated mode of thinking, which biases his thinking onwards; this severely constrains his possible theoretical point of views. The hope is then that the aforementioned self-correction of the science will lead to a healthier theoretical milieu, one in which a theorist with a different, more productive point of view, will have the opportunity to arise naturally as has happened in the past since Newton, without getting screamed down by proponents of a dominant competing theory to which he is forced to conform to from pretty much the start of his career or risk career suicide, as is the situation today.

 

[bhobba]

I wouldn't call multiverse interpretation "beautiful". Everett was kicked out of physics for proposing Many Worlds. The only reason multiverse models exist is because they are the only models that are consistent with quantum entanglement, Bell's Inequality, and non-simultaneity.

That's all wrong. Everett was not kicked out of physics, he was just the type of guy that liked solving problems - it didn't worry him that much what they were - he just decided to work on stuff for government instead.

The quantum formalism all by itself explains entanglement, Bell and all that. I have zero idea what you mean by non-simultaneity in QM. Standard QM does however obey the Galilean transformations.

Thanks
Bill

 

[gleem]

Didn't (doesn't) Lee Smolin also have a problem with the direction of theoretical physics in his book "The Trouble with Physics" Publish some eight years ago.

 

[gerald V]

I am surprised that Hossenfelder thinks that physics is mislead by the search for beauty. Actually, most of the current theories - inflation in particular - could hardly be more ugly. I have to accept such pragmatism, but hope that most of it will not be the last word.

To me it appears that physicist share the fate of all humans, namely that they are unable to learn from history. History of science teaches
that always those pre-judgements turned out as fallacious which had been regarded as most beyond dispute, supported by an intimate combination of experimental evidence and philosophical perception.

 

[Crass_Oscillator]

I find it baffling that anybody would suggest that beauty is anything but subjective. In quantum transport for instance, one can go the route of second quantization along with all of the associated dazzling Feynman diagrams and abstract representations, followed by conventional Green function methods, or for a number of other quantum kinetic approaches which are more crude or simple, where applicable (I confess that I like Wigner functions as, under the proper circumstances, I can look at the equation of motion cock-eyed until it looks like a Boltzmann equation, which is tractable to pleasing and comparatively simple Monte Carlo approaches).

By virtue of my different biases (applications to electronics in my case), my definition of beauty changed. If theoretical physicists are all clustering around the same biases (i.e. same standard of beauty) without progress in 40 years, Hossenfelder may have a point.

 

[bhobba]

I find it baffling that anybody would suggest that beauty is anything but subjective.

Of course in a statistical sense it is subjective ie there would be plenty of people who do not understand a theory well enough to really decide, they just have some pop sci idea of the theory or the general 'loose' ideas taught at HS, who would not say so, despite their source almost certainly saying it is.

The issue is however when you pick a genuinely beautiful theory such as GR, by which is meant just about all exposed to it's full technicalities, think GR is beautiful. In fact I do not know of anyone that does not (again of people that understand its technicalities). I think they generally hold views more along the lines of Chandrasekhar:
http://adsabs.harvard.edu/full/1984JApA...5...3C

That's the strange thing about theories generally considered beautiful. I would define a beautiful theory as one held as such, by an overwhelming majority of physicists, no physicist would care to argue it. Another characteristic is the theory is more or less inevitable from that beauty and is confirmed without revision by experiment.

Why some theories are like that I do not know, its a great mystery IMHO, but it seems to be the way some theories are.

Added Later:
One of the things I really enjoy is seeing a theory that on the surface looks ugly, what can be done to make it more like GR. I personally have nearly reached that point with QM - but not quite - close though.

Thanks
Bill

 

[fresh_42]

I find it baffling that anybody would suggest that beauty is anything but subjective.

I do. I find it baffling that anybody would suggest that beauty is something like taste. And yes, this does imply that not everybody can judge it. I find it exceptionally arrogant to assume that someone can actually judge on beauty regardless of what he knows. That is taste, not beauty.

We have a saying here: You cannot argue about taste. Either you have it or not.
That's an exaggeration, but it bears a germ of truth.

 

[Auto-Didact]

I find it baffling that anybody would suggest that beauty is anything but subjective. In quantum transport for instance, one can go the route of second quantization along with all of the associated dazzling Feynman diagrams and abstract representations, followed by conventional Green function methods, or for a number of other quantum kinetic approaches which are more crude or simple, where applicable (I confess that I like Wigner functions as, under the proper circumstances, I can look at the equation of motion cock-eyed until it looks like a Boltzmann equation, which is tractable to pleasing and comparatively simple Monte Carlo approaches).

By virtue of my different biases (applications to electronics in my case), my definition of beauty changed. If theoretical physicists are all clustering around the same biases (i.e. same standard of beauty) without progress in 40 years, Hossenfelder may have a point.

You have applications in mind, there's your problem ;) Hadamard and Poincaré, among others, have written on this exact topic.

Beauty in physics is an offshoot of beauty in pure mathematics. The branching off of the physicists perspective away from that of the mathematician occurred during the 19th century. Up until then physicists and a subset of mathematicians, mostly analysts and geometers, shared mostly the same standards of beauty.

Since then beauty in the eye of physicists has evolved in a few particular directions, while beauty in pure mathematics has split into several different opposing points of view: analysts, geometers, algebraists, logicists, formalists, and so on, each claiming superior beauty while simultaneously scoffing at that of the other.

 

[bhobba]

We have a saying here: You cannot argue about taste. Either you have it or not. That's an exaggeration, but it bears a germ of truth.

Not as much of an exaggeration as some may think. I used to be heavily into wine tasting and train people in how to do it. Many had never even tried to do it, and most in fact thought these great wines that you pay sometimes a bomb for is all hooey. Leaving aside the fact that some relatively cheap wines costing only $20,00 are in fact great, and a bottle of wine you can pick up here in Aus for $60.00 won the greatest wine in the world competition, wines with a great reputation and often because of that a very high price tag have that for very good reasons, and those reasons can be taught. The people were told what to look for eg examine color, nose, and palette separately and score using the international system. The palate must literally make your mouth juices flow - you should not be able to help it, it must reek of fruit, it must be in balance ie neither the fruit, acid or tannins should predominate, and of course it must delight you. Once given a bit of training you then do a proper blind wine tasting. Surprise surprise - they (as a group) pick the wines virtually in the same order as reputed quality from other tastings by professionals. Its sometimes a big shock. Then they become worried - how can I afford these wines - the others now taste so ordinary. A true wine lover knows to drink less but drink better.

Thanks
Bill

 

[Auto-Didact]

Not as much of an exaggeration as some may think. I used to be heavily into wine tasting and train people in how to do it. Many had never even tried to do it, and most in fact though these great wines that you pay sometimes a bomb for is all hooey. Leaving aside the fact that some relatively cheap wines costing only $20,00 are in fact great, and a bottle of wine you can pick up here in Aus for $60.00 won the greatest wine in the world competition, wines with a great reputation and often because of that a very high price tag have that for very good reasons, and those reasons can be taught. The people were told what to look for eg examine color, nose, and palette separately and score using the international system. The palate must literally make your mouth juices flow - you should not be able to help it, it must reek of fruit, it must be in balance ie neither the fruit, acid or tannins should predominate, and of course it must delight you. Once given a bit of training you then do a proper blind wine tasting. Surprise surprise - they (as a group) pick the wines virtually in the same order as reputed quality from other tastings by professionals. Its sometimes a big shock. Then they become worried - how can I afford these wines - the others now taste so ordinary. A true wine lover knows to drink less but drink better.

Thanks
Bill

Sounds exactly like expert training in medical diagnostics. One can definitely be trained to reliably identify certain signs and symptoms given proper medical history/physical examination training and sufficient exposure to physiological and pathological states; this isn't just my opinion, there is over 40 years of cognitive and psychological research on this. It is somewhat difficult to appreciate from the outside of medicine as one tends to get drowned in all the noise and small details, which tend to differ quite considerably between different specialties.

Its quite peculiar how different this kind of 'very precise in devilishly vague circumstances' reasoning feels in stark contrast to the more regular exact reasoning, i.e. the kind of reasoning predominantly used when solving problems in physics and mathematics which have readily known solutions and solving strategies. This is often labeled 'mathematical reasoning', but I think that is a misnomer as it refers mostly to simple problems solvable by basic (undergrad level) math skills.

However, with regards to the solving of problems without known solutions, without readily available strategies (e.g. nonlinear PDEs) and which are possibly incompletely stated, it becomes quite clear that there is quite a significant overlap with this 'non-mathematical' type of reasoning. In physics, it is exactly in this kind of reasoning that experts, i.e. theoretical and mathematical physicists, tend to be capable of reliably outperforming non-experts, i.e. other physicists and students alike. It's not merely a case of having more or different knowledge, theoreticians tend to reason in a different manner altogether.

 

[fresh_42]

Not as much of an exaggeration as some may think. I used to be heavily into wine tasting and train people in how to do it.

This reminds me of a story which once happened to me. I don't want to tell it, because it's a bit long, however, I found myself incidentally into a wine testing situation. I remember how I've been told about the difference of some wines coming from the same region: one from the north with mainly sandstone soil and one from the south with mainly limestone. What a surprise, that my untrained tongue could easily taste it just by looking for it - and no, it was definitely no psychological suggestion. It was rather obvious, I just hadn't been used to recognize it before. In the end I went home with a couple of pretty good wines, which weren't expensive at all. The guy who sold them simply knew all of his farmers personally and had wines from small farms, i.e. low distribution costs.

 

[bhobba]

I just hadn't been used to recognize it before.

And that is exactly where beauty in math/physics comes from. You can read in a book a theory or theorem is beautiful, but until you have seen the details yourself you can't appreciate it. And almost everyone that sees those details agrees.

Thanks
Bill

 

[marthasimons2]

There is nothing surprising about scientists looking for elegant and simple explanations. The theories that have already been proved are the evidence to the fact that our world's physical laws are beautiful. The point that we've spent a lot of money on proving some of the theories and were unlucky to do so does not mean that we have to stop trying. The point is to keep exploring our planet and the physical laws. If the question is in the distribution of wealth and that it would have been wiser to spend this money on something more productive, I could argue with this idea. We do not know what will be the results of the next scientific discovery. It might bring more use to the planet than anything that seems 'more practical'.

 

[ZapperZ]

Abraham Loeb wrote his view on this topic:

https://blogs.scientificamerican.co...sics-is-pointless-without-experimental-tests/

Zz.

 

[fresh_42]

There is nothing surprising about scientists looking for elegant and simple explanations. The theories that have already been proved are the evidence to the fact that our world's physical laws are beautiful. The point that we've spent a lot of money on proving some of the theories and were unlucky to do so does not mean that we have to stop trying. The point is to keep exploring our planet and the physical laws. If the question is in the distribution of wealth and that it would have been wiser to spend this money on something more productive, I could argue with this idea. We do not know what will be the results of the next scientific discovery. It might bring more use to the planet than anything that seems 'more practical'.

Oh, there come a whole lot of more things to mind, which I think we could easily drop and save money. Big money.

 

[gleem]

George Will a political columnist wrote an unusual article IMO about basic research and it may not be what you think.

https://www.washingtonpost.com/opin...0e80e1fdf43_story.html?utm_term=.adb563395e7f

 

[Crass_Oscillator]

Obviously if you begin with the same values, you will converge to an "objective" notion of beauty. For instance, I highly value parsimony, and, therefore, find GR to be aesthetically pleasing, even though in practice it is mathematically turgid. Some of you probably do too.

The problem is that some value systems are degenerate, in multiple senses of the word, such as the value system that concludes convoluted abstraction is automatically beautiful. These are the folks who are attracted to topological phases or string theory like mosquitoes to lanterns, and they often claim to appreciate GR for utterly the wrong reasons, which is how they market supersymmetry and other distractions.

I'm no idol worshiper but I highly doubt Einstein's aesthetic preferences would have tolerated something like M-theory.

 

[ZapperZ]

Sabine Hossenfelder's talk at BNL is now online.



Zz.

 

[Lish Lash]

Here's a direct link to one of Hossenfelder's points most relevant to this discussion, "what physicists mean when they talk about beauty":